Math Problem Statement

The band is breaking up and Rob, Sue, Tim and Vito each want the tour bus. Using the method of sealed bids, Rob bids $4300, Sue bids $5900, Tim bids $3500, and Vito bids $6200 for the bus. Since Vito's bid is the highest, he gets the bus and has to compensate the others to make the division fair. How much money will Rob get from Vito? Be sure to compute your answer to the nearest cent.

Solution

To determine how much Rob, Sue, and Tim should get from Vito using the method of sealed bids, we will follow these steps:

Step 1: Calculate the total of all the bids.

The total bids from all four individuals are: 4300+5900+3500+6200=199004300 + 5900 + 3500 + 6200 = 19900

Step 2: Calculate the fair share for each person.

The fair share for each person is computed by dividing the total bid by the number of people (4): Fair share for each person=199004=4975\text{Fair share for each person} = \frac{19900}{4} = 4975

Step 3: Determine how much Vito must pay.

Vito's bid for the bus was $6200, which is more than his fair share of $4975. Therefore, he must compensate the other three to make the division fair.

Step 4: Compute the compensation to Rob.

Since Vito's bid exceeded his fair share by: 62004975=12256200 - 4975 = 1225 Vito must compensate Rob, Sue, and Tim by distributing this amount in proportion to their fair shares.

To determine how much Rob will receive, we subtract his fair share from his bid: Rob’s bid=4300,Rob’s fair share=4975\text{Rob's bid} = 4300, \quad \text{Rob's fair share} = 4975 Rob's shortfall is: 49754300=6754975 - 4300 = 675

Step 5: Vito compensates Rob.

Vito will give Rob the shortfall amount, which is: 675 dollars\boxed{675 \text{ dollars}}

Thus, Rob will receive $675 from Vito.

Would you like further details or have any questions? Here are five related questions you might find interesting:

  1. How much will Sue receive from Vito?
  2. What is Tim's compensation in this scenario?
  3. What would happen if someone bids less than their fair share?
  4. How can we ensure that the method of sealed bids is fair for everyone involved?
  5. How is this method applied in other real-world situations like estate divisions?

Tip: The sealed bids method is often used to fairly divide assets when people have different valuations of them!

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Math Problem Analysis

Mathematical Concepts

Sealed Bids Method
Fair Division
Proportional Distribution

Formulas

Total Bids = Sum of all individual bids
Fair Share = Total Bids / Number of participants
Compensation = Fair Share - Individual Bid

Theorems

Fair Division Principle

Suitable Grade Level

Grades 9-12